Formulas

In the following, is an approximation in the frame where the initial electron and laser collide head-on and the electron is ultra-relativistic.

p,,p',k
4-momenta of initial electron, laser photon, final electron and emitted photon, respectively.
,,,
Energies of initial electron, laser photon, final electron and emitted photon, respectively.

Laser energy parameter: .
n
Number of absorbed laser photon. The kinetic relation holds exactly. Here, q is defined as

(p is replaced by p' for q') and is called quasimomentum.
x
, (0<x<1)
v
v=x/(1-x), x=v/(1+v). () .

Maximum v for given n: .

Maximum x for given n: .

Laser helicity (-1 or +1)
,
Initial and final electron helicities ()

Final photon helicity
,
`Detector helicity' of the final particles. See section 65 of [3].

is the effective energy of initial electron in the laser field.

Final photon angle.


The argument of the Bessel functions in the following expressions:

Number of photons per unit time is
 
The terms involving and simultaneously are ignored, i.e., the correlation of polarization between final particles is ignored. The ultra-relativistic approximation has been applied in the terms related to electron helicity ( and/or ). (Note that the electron helicity is a Lorentz invariant quantity only in the ultra-relativistic limit.)

The sum over the final electron and photon helicities gives
 
The functions are defined by

, , , are identical to , , , divided by in Tsai's paper[5], although the expressions in his paper look much more complicated.

Once x and n are given, the final momentuma are given, in any frame, by



Here, is the azimuthal angle in a head-on frame (therefore its distribution is uniform in [0,]) and and are given by
 
where is the completely anti-symmetric tensor (). These vectors satisfy

The vector in eq.(139) is ill-defined when and are colinear in the original frame. In such a case the spatial part of is an arbitrary unit vector perpendicular to .